GeneralizedPareto {actuar}R Documentation

The Generalized Pareto Distribution

Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Generalized Pareto distribution with parameters shape1, shape2 and scale.

Usage

dgenpareto(x, shape1, shape2, rate = 1, scale = 1/rate,
           log = FALSE)
pgenpareto(q, shape1, shape2, rate = 1, scale = 1/rate,
           lower.tail = TRUE, log.p = FALSE)
qgenpareto(p, shape1, shape2, rate = 1, scale = 1/rate,
           lower.tail = TRUE, log.p = FALSE)
rgenpareto(n, shape1, shape2, rate = 1, scale = 1/rate)
mgenpareto(order, shape1, shape2, rate = 1, scale = 1/rate)
levgenpareto(limit, shape1, shape2, rate = 1, scale = 1/rate,
             order = 1)

Arguments

x, q vector of quantiles.
p vector of probabilities.
n number of observations. If length(n) > 1, the length is taken to be the number required.
shape1, shape2, scale parameters. Must be strictly positive.
rate an alternative way to specify the scale.
log, log.p logical; if TRUE, probabilities/densities p are returned as log(p).
lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
order order of the moment.
limit limit of the loss variable.

Details

The Generalized Pareto distribution with parameters shape1 = a, shape2 = b and scale = s has density:

f(x) = Gamma(a + b)/(Gamma(a) * Gamma(b)) (s^a x^(b - 1))/ (x + s)^(a + b)

for x > 0, a > 0, b > 0 and s > 0. (Here Gamma(a) is the function implemented by R's gamma() and defined in its help.)

The Generalized Pareto is the distribution of the random variable

s (X/(1 - X)),

where X has a Beta distribution with parameters a and b.

The Generalized Pareto distribution has the following special cases:

Value

dgenpareto gives the density, pgenpareto gives the distribution function, qgenpareto gives the quantile function, rgenpareto generates random deviates, mgenpareto gives the kth raw moment, and levgenpareto gives the kth moment of the limited loss variable.
Invalid arguments will result in return value NaN, with a warning.

Note

Distribution also known as the Beta of the Second Kind.

Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004), Loss Models, From Data to Decisions, Second Edition, Wiley.

Examples

exp(dgenpareto(3, 3, 4, 4, log = TRUE))
p <- (1:10)/10
pgenpareto(qgenpareto(p, 3, 3, 1), 3, 3, 1)
qgenpareto(.3, 3, 4, 4, lower.tail = FALSE)
mgenpareto(1, 3, 2, 1) ^ 2
levgenpareto(10, 3, 3, 3, order = 2)
[Package actuar version 1.0-2 Index]